Logistic Regression

library(tidyverse)
library(knitr)
library(plotly) ; library(viridis) ; library(gridExtra) ; library(RColorBrewer)
library(biomaRt)
library(Rtsne)
library(ROCR) ; library(car)
library(corrplot)
library(expss) ; library(knitr)

SFARI_colour_hue = function(r) {
  pal = c('#FF7631','#FFB100','#E8E328','#8CC83F','#62CCA6','#59B9C9','#b3b3b3','#808080','gray','#d9d9d9')[r]
}

Load and prepare data

Load dataset (preprocessing code in 20_04_07_create_dataset.html)

clustering_selected = 'DynamicHybridMergedSmall'
print(paste0('Using clustering ', clustering_selected))

## [1] "Using clustering DynamicHybridMergedSmall"

# Dataset created with DynamicTreeMerged algorithm
dataset = read.csv(paste0('./../Data/dataset_', clustering_selected, '.csv'), row.names=1)

# Add gene symbol
getinfo = c('ensembl_gene_id','external_gene_id')
mart = useMart(biomart='ENSEMBL_MART_ENSEMBL', dataset='hsapiens_gene_ensembl',
               host='feb2014.archive.ensembl.org') ## Gencode v19
gene_names = getBM(attributes=getinfo, filters=c('ensembl_gene_id'), values=rownames(dataset), mart=mart)

rm(getinfo, mart)

Gene filtering:

• Remove genes without cluster (Module=gray)

rm_cluster = dataset[is.na(dataset$MTcor),'Module'] %>% unique %>% as.character

print(paste0('Removing ', sum(dataset$Module=='gray'), ' genes without cluster'))

## [1] "Removing 113 genes without cluster"

new_dataset = dataset %>% filter(Module != 'gray' & !is.na(MTcor))

Variable changes:

• Using Module Membership variables instead of binary module membership

• Not including p-value variables

• Including a new variable with the absolute value of GS

• Removing information from gray module (unclassified genes)

• Objective variable: Binary label indicating if it’s in the SFARI dataset or not

new_dataset = new_dataset %>% dplyr::select(-c(matches(paste('pval|Module')), MMgray)) %>%
              mutate('absGS'=abs(GS), 'SFARI'=ifelse(gene.score=='None', FALSE, TRUE)) %>%
              dplyr::select(-gene.score)

rownames(new_dataset) = rownames(dataset)[dataset$Module != 'gray']

rm(rm_cluster)

original_dataset = dataset
dataset = new_dataset
print(paste0('The final dataset contains ', nrow(dataset), ' observations and ', ncol(dataset), ' variables'))

## [1] "The final dataset contains 16034 observations and 34 variables"

rm(new_dataset)

Sample of the dataset (transposed so it’s easier to see)

dataset %>% head(5) %>% t %>% kable

	ENSG00000000003	ENSG00000000419	ENSG00000000457	ENSG00000000460	ENSG00000000938
MTcor	0.5704012	0.2525473	-0.9514071	-0.8039747	0.8771832
GS	0.3866600	0.0319742	-0.3948272	-0.2425891	0.4206897
MM.B79F00	-0.3421536	-0.0685993	0.1822384	-0.0203122	0.0401878
MM.FE6E8A	-0.0815013	-0.2406456	-0.0314884	-0.1149264	0.2532996
MM.00BF7D	-0.1457984	-0.1570793	0.2895552	0.2146387	-0.1582971
MM.D376FF	-0.1558365	-0.3275203	0.2216473	0.1076922	-0.1267206
MM.00B7E9	-0.2706076	0.1086927	0.2716955	0.3287583	-0.1127668
MM.00BD5F	-0.2103137	-0.0739132	0.1588410	0.1164722	-0.1165457
MM.00BA38	-0.2073096	-0.2063582	0.1423063	-0.0137954	-0.2407892
MM.FF62BC	0.0881871	-0.0729658	-0.0536479	-0.1519834	0.0208935
MM.619CFF	-0.0278009	-0.3888226	0.1529542	0.0699100	-0.0311339
MM.EF7F49	-0.0743495	-0.3953383	0.1857405	-0.1853277	-0.0511009
MM.F8766D	0.0320566	-0.2755247	-0.0473936	-0.2555324	-0.0717451
MM.00C0AF	-0.3240023	-0.0804812	0.4041119	0.1597938	-0.4479131
MM.B983FF	-0.1398423	-0.1306399	0.2955196	0.0852169	-0.3261794
MM.F564E3	-0.0472562	-0.0021517	0.1666066	-0.0218248	-0.2134750
MM.FD61D1	-0.2102549	0.3544128	0.2690348	0.4502318	-0.4636712
MM.8AAB00	0.1951790	0.3820330	-0.0471786	0.2444100	-0.1238171
MM.C99800	0.1463015	0.5008550	-0.0892625	0.3573500	-0.0912000
MM.00BFC4	-0.0419816	0.0223852	-0.0618101	-0.0495563	0.2941168
MM.E58700	0.1436711	-0.0157220	-0.3400724	-0.2724479	0.1215078
MM.FF67A4	0.2300837	-0.1215257	-0.1906585	-0.3770794	0.1655206
MM.00A7FF	-0.1508215	0.1640578	0.0744979	0.1972789	-0.1825727
MM.6BB100	-0.1412895	0.0441975	0.0625034	0.2417779	0.0799198
MM.00C097	0.1450441	0.0257528	-0.2382952	-0.0953965	0.2989640
MM.39B600	0.3292611	-0.0759346	-0.3095235	-0.3958098	0.2579894
MM.9590FF	0.2281574	0.1383876	-0.2008810	0.0354509	0.3490105
MM.A3A500	0.2008548	0.0853587	-0.1824296	-0.1208052	0.2633420
MM.00BCD8	0.0804575	-0.0263322	-0.2410231	0.0801394	0.0722277
MM.E76BF3	0.4768465	-0.1831555	-0.1360703	-0.2163761	0.1917615
MM.00B0F6	0.3884997	0.1582543	-0.3053931	-0.2093428	0.3554336
MM.D89000	0.1285911	0.1002678	-0.1304555	0.1913294	0.0942191
absGS	0.3866600	0.0319742	0.3948272	0.2425891	0.4206897
SFARI	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000

Objective variable distribution: Unbalanced labels

print(table(dataset$SFARI))

## 
## FALSE  TRUE 
## 15137   897

cat(paste0('\n',round(mean(dataset$SFARI)*100,2), '% of the observations are positive'))

## 
## 5.59% of the observations are positive

---

Visualisations

Visualising the variables

Chose the t-SNE algorithm because it preserves distances

The SFARI labels is still close to the absolute value of Gene Significance. This time the MM variables seem to be grouped in 2 clusters

tsne = dataset %>% t %>% Rtsne(perplexity=10)

plot_data = data.frame('ID'=colnames(dataset), 'C1'=tsne$Y[,1], 'C2'=tsne$Y[,2],
                       type=ifelse(grepl('MM', colnames(dataset)),'ModMembership',
                            ifelse(grepl('SFARI', colnames(dataset)), 'SFARI',
                            ifelse(grepl('GS', colnames(dataset)), 'GS', 'MTcor'))))

ggplotly(plot_data %>% ggplot(aes(C1, C2, color=type)) + geom_point(aes(id=ID)) + 
         theme_minimal() + ggtitle('t-SNE visualisation of variables'))

The Module Membership variables are grouped by Module-Trait correlation, with positive correlations on one side, negative on the other, and modules with low correlation far away from the SFARI tag

mtcor_by_module = original_dataset %>% dplyr::select(Module, MTcor) %>% unique
colnames(mtcor_by_module) = c('ID','MTcor')

plot_data = mtcor_by_module %>% mutate(ID = gsub('#','MM.',ID)) %>% right_join(plot_data, by='ID')

ggplotly(plot_data %>% ggplot(aes(C1, C2, color=MTcor)) + geom_point(aes(id=ID)) + 
         scale_color_viridis() + theme_minimal() + 
         ggtitle('t-SNE of variables coloured by Module-Diagnosis correlation'))

rm(mtcor_by_module, tsne)

Visualising the observations

• The two main patterns that seem to characterise the genes are their Gene Significance and the Module-Diagnosis correlation of their corresponding module

• Mean Expression doesn’t seem to play an important role

• I don’t know what the 2nd PC is capturing

# Mean Expression data
load('./../Data/preprocessed_data.RData')

## Registered S3 methods overwritten by 'Hmisc':
##   method                 from 
##   [.labelled             expss
##   print.labelled         expss
##   as.data.frame.labelled expss

datExpr = datExpr %>% data.frame
mean_expr = data.frame('ID'=rownames(datExpr), 'meanExpr' = rowMeans(datExpr))

# PCA
pca = dataset %>% t %>% prcomp

plot_data = data.frame('ID'=rownames(dataset), 'PC1'=pca$rotation[,1], 'PC2'=pca$rotation[,2], 
                       'SFARI'=dataset$SFARI, 'MTcor'=dataset$MTcor, 'GS'=dataset$GS) %>%
            mutate(alpha=ifelse(SFARI, 0.7, 0.2)) %>% left_join(mean_expr, by='ID')

p1 = plot_data %>% ggplot(aes(PC1, PC2, color=MTcor)) + geom_point(alpha=0.4) + scale_color_viridis() + 
     theme_minimal() + ggtitle('Genes coloured by Module-Diagnosis correlation') +
     xlab(paste0('PC1 (',round(100*summary(pca)$importance[2,1]),'%)')) +
     ylab(paste0('PC2 (',round(100*summary(pca)$importance[2,2]),'%)')) +
     theme(legend.position='bottom')

p2 = plot_data %>% ggplot(aes(PC1, PC2, color=GS)) + geom_point(alpha=0.4) + scale_color_viridis() + 
     theme_minimal() + ggtitle('Genes coloured by Gene Significance') + theme(legend.position='bottom')

p3 = plot_data %>% ggplot(aes(PC1, PC2, color=SFARI)) + geom_point(alpha = plot_data$alpha) +
     theme_minimal() + ggtitle('Genes coloured by SFARI label') + theme(legend.position='bottom')
p3 = ggExtra::ggMarginal(p3, type='density', groupColour=TRUE, size=10)

p4 = plot_data %>% ggplot(aes(PC1, PC2, color=meanExpr)) + geom_point(alpha=0.4) + scale_color_viridis() + 
     theme_minimal() + ggtitle('Genes coloured by mean level of expression') + theme(legend.position='bottom')

grid.arrange(p1, p2, p3, p4, nrow=2)

rm(pca, datExpr, datGenes, datMeta, dds, DE_info, mean_expr, p1, p2, p3, p4)

---

Resampling to reduce class imbalance

For now, will do this using over- and under-sampling of the classes, but later on should check SMOTE (Synthetic Minority Over-sampling Technique) method

Need to divide first into train and test sets to keep the sets independent: using 80% of the Positive observations on the training set

Note: Even though our label is binary, I want to have representative samples for all SFARI scores in both the training and test data, so instead of pooling all the SFARI scores together and randomly selecting 80% of the samples, I’m going to create the positive set selecting 80% of each of the samples by score

set.seed(123)

positive_sample_balancing_SFARI_scores = function(p){
  
  positive_train_idx = c()
  positive_test_idx = c()
  
  for(score in 1:6){
    score_genes = rownames(original_dataset)[rownames(original_dataset) %in% rownames(dataset) & original_dataset$gene.score == score]
    score_idx = which(rownames(dataset) %in% score_genes)
    score_train_idx = sample(score_idx, size = ceiling(p*length(score_idx)))
    score_test_idx = score_idx[!score_idx %in% score_train_idx]
    
    positive_train_idx = c(positive_train_idx, score_train_idx)
    positive_test_idx = c(positive_test_idx, score_test_idx) 
  }
  
  return(list('train' = sort(positive_train_idx), 'test' = sort(positive_test_idx)))
}

# 80% of the samples for the training set
p = 0.8

positive_idx = positive_sample_balancing_SFARI_scores(p)
positive_train_idx = positive_idx[['train']]
positive_test_idx = positive_idx[['test']]

negative_idx = which(!dataset$SFARI)
negative_train_idx = sort(sample(negative_idx, size=ceiling(p*length(negative_idx))))
negative_test_idx = negative_idx[!negative_idx %in% negative_train_idx] # This is overwritten below because of the subbsampling

train_set = dataset[c(positive_train_idx, negative_train_idx),]
test_set = dataset[c(positive_test_idx, negative_test_idx),] # This is overwritten below because of the subbsampling

rm(positive_idx, negative_idx, positive_train_idx, positive_test_idx, negative_train_idx, negative_test_idx,
   p, positive_sample_balancing_SFARI_scores)

Balancing the dataset to obtain a 1:1 ratio in labels

Over-sampling observations with positive SFARI label: Sample with replacement 4x original number of observations

Sample with replacement positive observations in train set

positive_obs = which(train_set$SFARI)

add_obs = sample(positive_obs, size=3*length(positive_obs), replace=TRUE)

train_set = train_set[c(1:nrow(train_set), add_obs),]

rm(positive_obs, add_obs)

Under-sampling observations with negative SFARI labels

cat(paste0('Keeping ~',round(100*sum(train_set$SFARI)/sum(!train_set$SFARI)),
             '% of the Negative observations in the training set'))

## Keeping ~24% of the Negative observations in the training set

negative_obs = which(!train_set$SFARI)

keep_obs = sample(negative_obs, size=sum(train_set$SFARI))

train_set = train_set[c(keep_obs, which(train_set$SFARI)),]

# Add observations we removed from the training data to the test data
lost_obs = !rownames(dataset) %in% c(rownames(train_set), rownames(test_set))
cat(paste0('Adding ', sum(lost_obs), ' Negative samples to the test set from the ones we just removed from the training set'))

## Adding 9230 Negative samples to the test set from the ones we just removed from the training set

test_set = rbind(test_set, dataset[lost_obs,])


rm(negative_obs, keep_obs)

Label distribution in training set

cro(train_set$SFARI)

	#Total
train_set$SFARI
FALSE	2880
TRUE	2880
#Total cases	5760

Labels distribution in test set

cro(test_set$SFARI)

	#Total
test_set$SFARI
FALSE	12257
TRUE	177
#Total cases	12434

---

Logistic Regression

Train model

train_set$SFARI = train_set$SFARI %>% as.factor

fit = glm(SFARI~., data=train_set, family='binomial')

Predict labels in test set

test_set$prob = predict(fit, newdata=test_set, type='response')
test_set$pred = test_set$prob>0.5

Performance metrics

Confusion matrix

conf_mat = test_set %>% apply_labels(SFARI = 'Actual Labels', 
                                     prob = 'Assigned Probability', 
                                     pred = 'Label Prediction')

cro(conf_mat$SFARI, list(conf_mat$pred, total()))

	Label Prediction			#Total
	FALSE	TRUE
Actual Labels
FALSE	8091	4166		12257
TRUE	76	101		177
#Total cases	8167	4267		12434

rm(conf_mat)

Accuracy

acc = mean(test_set$SFARI==test_set$pred)

cat(paste0('Accuracy = ', round(acc,4)))

## Accuracy = 0.6588

rm(acc)

ROC Curve

pred_ROCR = prediction(test_set$prob, test_set$SFARI)

roc_ROCR = performance(pred_ROCR, measure='tpr', x.measure='fpr')
AUC = performance(pred_ROCR, measure='auc')@y.values[[1]]

plot(roc_ROCR, main=paste0('ROC curve (AUC=',round(AUC,2),')'), col='#009999')
abline(a=0, b=1, col='#666666')

Lift Curve

lift_ROCR = performance(pred_ROCR, measure='lift', x.measure='rpp')
plot(lift_ROCR, main='Lift curve', col='#86b300')

rm(pred_ROCR, roc_ROCR, AUC, lift_ROCR)

---

Analyse model

SFARI genes have a slightly higher score distribution than the rest

plot_data = test_set %>% dplyr::select(prob, SFARI)

ggplotly(plot_data %>% ggplot(aes(prob, fill=SFARI, color=SFARI)) + geom_density(alpha=0.3) + xlab('Score') +
         geom_vline(xintercept = mean(plot_data$prob[plot_data$SFARI]), color = '#00C0C2', linetype = 'dashed') +
         geom_vline(xintercept = mean(plot_data$prob[!plot_data$SFARI]), color = '#FF7371', linetype = 'dashed') +
         theme_minimal() + ggtitle('Model score distribution by SFARI Label'))

• There seems to be a positie relation between the SFARI scores and the probability assigned by the model

• The number of observations when separating the test set by SFARI score is quite small, so this is not a robust result, specially for scores 1, 2 and 6

plot_data = test_set %>% mutate(ID=rownames(test_set)) %>% dplyr::select(ID, prob) %>%
            left_join(original_dataset %>% mutate(ID=rownames(original_dataset)), by='ID') %>%
            dplyr::select(ID, prob, gene.score) %>% apply_labels(gene.score='SFARI Gene score')

cro(plot_data$gene.score)

	#Total
SFARI Gene score
1	5
2	12
3	38
4	86
5	32
6	4
None	12257
#Total cases	12434

mean_vals = plot_data %>% group_by(gene.score) %>% summarise(mean_prob = mean(prob))

# plot_data %>% ggplot(aes(prob, color=gene.score, fill=gene.score)) + geom_density(alpha=0.25) + 
#               geom_vline(data=mean_vals, aes(xintercept=mean_prob, color=gene.score), linetype='dashed') +
#               scale_colour_manual(values=SFARI_colour_hue(r=c(1:6,8,7))) +
#               scale_fill_manual(values=SFARI_colour_hue(r=c(1:6,8,7))) + 
#               ggtitle('Distribution of probabilities by SFARI score') +
#               xlab('Probability') + ylab('Density') + theme_minimal()

ggplotly(plot_data %>% ggplot(aes(gene.score, prob, fill=gene.score)) + geom_boxplot() + 
              scale_fill_manual(values=SFARI_colour_hue(r=c(1:6,8,7))) + 
              ggtitle('Distribution of probabilities by SFARI score') +
              xlab('SFARI score') + ylab('Probability') + theme_minimal())

rm(mean_vals)

Testing for Multicollinearity with Variance Inflation Factors (VIF)

Any variable with a VIF above 10 is considered to have strong multicollinearity: the dataset has a really big problem with multicollinearity :/

Multicollinearity affects the coefficients and p-values of the regression, but it doesn’t affect the predictions, precision of the predictions or the goodness-of-fit statistics ref. So we cannot analyse the coefficients to see which features are the most important, but all the results from this analysis are valid

# VIF
plot_data = data.frame('Feature' = car::vif(fit) %>% sort %>% names,
                       'VIF' = car::vif(fit) %>% sort %>% unname) %>%
            mutate(outlier = VIF>10)

plot_data %>% ggplot(aes(reorder(Feature, -VIF), VIF, fill = !outlier)) + geom_bar(stat='identity') + scale_y_log10() +
              geom_hline(yintercept = 10, color = 'gray', linetype = 'dashed') + xlab('Model Features') + theme_minimal() +
              theme(legend.position = 'none', axis.text.x = element_text(angle = 90, hjust = 1))

# Correlation plot
corrplot.mixed(cor(train_set[,-ncol(train_set)]), lower = 'number', lower.col = 'gray', number.cex = .6, tl.pos = 'l', tl.col = '#666666')

Genes with highest scores in test set

• Considering the imbalance in labels in the test set (1:69), there is a high concentration of SFARI scores in the top genes (1:13)

• 3 of the 5 SFARI genes with a score of 1 in the test set are in this top 50 list

• There aren’t any SFARI genes with a score lower than 3 (even though 69% of the SFARI genes in the test score have a score lower than 3)

test_set %>% dplyr::select(prob, SFARI) %>% mutate(ID = rownames(test_set)) %>% 
             arrange(desc(prob)) %>% top_n(50, wt=prob) %>%
             left_join(original_dataset %>% mutate(ID=rownames(original_dataset)), by='ID')  %>% 
             left_join(gene_names, by = c('ID'='ensembl_gene_id')) %>%
             dplyr::rename('GeneSymbol' = external_gene_id, 'Probability' = prob, 'ModuleDiagnosis_corr' = MTcor) %>%
             mutate(ModuleDiagnosis_corr = round(ModuleDiagnosis_corr,4), Probability = round(Probability,4)) %>%
             dplyr::select(GeneSymbol, gene.score, ModuleDiagnosis_corr, Module, Probability) %>%
             kable(caption = 'Genes with highest model probabilities from the test set')

Genes with highest model probabilities from the test set

GeneSymbol	gene.score	ModuleDiagnosis_corr	Module	Probability
RPRD2	None	-0.6031	#00BA38	0.8966
MYCBP2	None	-0.6031	#00BA38	0.8907
KCNJ6	None	-0.9514	#00C0AF	0.8815
AAK1	None	-0.9514	#00C0AF	0.8704
ARID1B	1	0.1127	#FF62BC	0.8693
RASGRF1	None	-0.9514	#00C0AF	0.8658
RFX7	None	0.0586	#FE6E8A	0.8636
SAP130	None	-0.6031	#00BA38	0.8612
FMNL1	None	-0.6031	#00BA38	0.8605
HUNK	None	-0.6750	#D376FF	0.8557
PAK3	None	-0.9514	#00C0AF	0.8555
ST8SIA3	None	-0.9514	#00C0AF	0.8542
NMNAT2	None	-0.9514	#00C0AF	0.8538
KMT2D	None	-0.6031	#00BA38	0.8534
PCLO	None	0.0586	#FE6E8A	0.8523
CACNG3	None	-0.6031	#00BA38	0.8522
TLN2	None	-0.9514	#00C0AF	0.8516
FAM193A	None	0.0586	#FE6E8A	0.8492
CELF2	None	-0.9514	#00C0AF	0.8491
TSPYL4	None	-0.9514	#00C0AF	0.8473
NAV1	None	-0.6031	#00BA38	0.8464
CLIP3	None	-0.6031	#00BA38	0.8459
KMT2A	1	0.7916	#00C097	0.8447
MIDN	None	-0.6031	#00BA38	0.8441
PDS5B	None	-0.9514	#00C0AF	0.8424
RNF111	None	0.0586	#FE6E8A	0.8421
SCN2A	1	-0.9514	#00C0AF	0.8416
EIF4G3	None	-0.8040	#00B7E9	0.8383
EP400	3	-0.6031	#00BA38	0.8366
BAZ2A	None	-0.0094	#00A7FF	0.8364
ATN1	None	-0.6031	#00BA38	0.8354
GATAD2B	None	-0.6031	#00BA38	0.8352
POM121C	None	-0.6031	#00BA38	0.8348
ASAP1	None	-0.6031	#00BA38	0.8344
PCDHB13	None	-0.4946	#B79F00	0.8342
BICD1	None	0.0586	#FE6E8A	0.8340
SMG7	None	-0.6031	#00BA38	0.8339
AFF3	None	0.0586	#FE6E8A	0.8324
UBR4	None	-0.6031	#00BA38	0.8322
KIAA0226	None	-0.6031	#00BA38	0.8312
HIVEP2	None	-0.9514	#00C0AF	0.8310
PEG3	None	-0.9514	#00C0AF	0.8307
ARHGEF3	None	-0.9514	#00C0AF	0.8306
BRPF3	None	-0.6031	#00BA38	0.8280
SCN3B	None	0.1127	#FF62BC	0.8276
PATL1	None	0.1127	#FF62BC	0.8269
RNF150	None	-0.0094	#00A7FF	0.8266
RIMBP2	None	-0.0094	#00A7FF	0.8265
DAAM1	None	-0.0094	#00A7FF	0.8259
CHAMP1	None	-0.6031	#00BA38	0.8254

Negative samples distribution

Running the model on all non-SFARI genes (excluding the ones in the train set)

negative_set = dataset %>% filter(!SFARI & !rownames(.) %in% rownames(train_set)) %>% dplyr::select(-SFARI)
rownames(negative_set) = rownames(dataset)[!dataset$SFARI & !rownames(dataset) %in% rownames(train_set)]

negative_set$prob = predict(fit, newdata=negative_set, type='response')
negative_set$pred = negative_set$prob>0.5

negative_set_table = negative_set %>% apply_labels(prob = 'Assigned Probability', 
                                                   pred = 'Label Prediction')

cro(negative_set_table$pred)

	#Total
Label Prediction
FALSE	8091
TRUE	4166
#Total cases	12257

cat(paste0('\n', sum(negative_set$pred), ' genes are predicted as ASD-related'))

## 
## 4166 genes are predicted as ASD-related

negative_set %>% ggplot(aes(prob)) + geom_density(color='#F8766D', fill='#F8766D', alpha=0.5) +
                 geom_vline(xintercept=0.5, color='#333333', linetype='dotted') + 
                 ggtitle('Probability distribution of all the Negative samples in the dataset') + 
                 theme_minimal()

Probability and Gene Significance

There’s a lot of noise, but the genes with the highest probabilities have slightly higher (absolute) Gene Significance

negative_set %>% ggplot(aes(prob, GS, color=MTcor)) + geom_point() + geom_smooth(method='loess', color='#666666') +
                 geom_hline(yintercept=0, color='gray', linetype='dashed') +
                 scale_color_gradientn(colours=c('#F8766D','white','#00BFC4')) + 
                 ggtitle('Relation between Probability and Gene Significance') + theme_minimal()

negative_set %>% ggplot(aes(prob, abs(GS), color=MTcor)) + geom_point() + 
                 geom_hline(yintercept=mean(negative_set$absGS), color='gray', linetype='dashed') + 
                 geom_smooth(method='loess', color='#666666') +
                 scale_color_gradientn(colours=c('#F8766D','white','#00BFC4')) + 
                 ggtitle('Relation between Model probability and Gene Significance') + theme_minimal()

Probability and Module-Diagnosis correlation

On average, the model seems to be assigning a probability inversely proportional to the Module-Diagnosis correlation of the module, with the highest positively correlated modules having the lowest average probability and the highest negatively correlated modules the highest average probability. But the difference isn’t big

negative_set %>% ggplot(aes(MTcor, prob, color=GS)) + geom_point() + geom_smooth(method='loess', color='#666666') + 
                 geom_hline(yintercept=mean(negative_set$prob), color='gray', linetype='dashed') +
                 scale_color_gradientn(colours=c('#F8766D','#F8766D','white','#00BFC4','#00BFC4')) + 
                 xlab('Modules ordered by their correlation to ASD') + ylab('Model probability') +
                 theme_minimal()

Summarised version, plotting by module instead of by gene

The difference in the trend lines between this plot and the one above is that the one above takes all the points into consideration while this considers each module as an observation by itself, so the top one is strongly affected by big modules and the bottom one treats all modules the same

The model seems to give higher probabilities to genes belonging to modules with a small (absolute) correlation to Diagnosis, although the difference isn’t much

plot_data = negative_set %>% group_by(MTcor) %>% summarise(mean = mean(prob), sd = sd(prob), n = n()) %>%
            mutate(MTcor_sign = ifelse(MTcor>0, 'Positive', 'Negative')) %>% left_join(original_dataset, by='MTcor') %>%
            dplyr::select(Module, MTcor, MTcor_sign, mean, sd, n) %>% distinct()
colnames(plot_data)[1] = 'ID'

ggplotly(plot_data %>% ggplot(aes(MTcor, mean, size=n, color=MTcor_sign)) + geom_point(aes(id=ID)) + 
         geom_smooth(method='loess', color='gray', se=FALSE) + geom_smooth(method='lm', se=FALSE) + 
         xlab('Module-Diagnosis correlation') + ylab('Mean Probability by Model') + theme_minimal())

Probability and mean level of expression

There is a positive relation between level of expression and probability, the model seems to be capturing indirectly the level of expression of the genes to make the prediction, so it’s introducing the same bias

# Gandal dataset
load('./../Data/preprocessed_data.RData')
datExpr = datExpr %>% data.frame
DE_info = DE_info %>% data.frame

mean_and_sd = data.frame(ID=rownames(datExpr), meanExpr=rowMeans(datExpr), sdExpr=apply(datExpr,1,sd))

plot_data = negative_set %>% mutate(ID=rownames(negative_set)) %>% left_join(mean_and_sd, by='ID') %>% 
            left_join(original_dataset %>% mutate(ID=rownames(original_dataset)) %>% 
                      dplyr::select(ID, Module), by='ID')
colnames(plot_data)[ncol(plot_data)] = 'Module'

plot_data %>% ggplot(aes(meanExpr, prob)) + geom_point(alpha=0.2, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.3) + 
              geom_smooth(method='lm', color='#999999', se=FALSE, alpha=1) + 
              theme_minimal() + ggtitle('Mean expression vs model probability by gene')

rm(mean_and_sd)

plot_data2 = plot_data %>% group_by(Module) %>% summarise(meanExpr = mean(meanExpr), meanProb = mean(prob), n=n())

ggplotly(plot_data2 %>% ggplot(aes(meanExpr, meanProb, size=n)) + geom_point(color=plot_data2$Module) + 
         geom_smooth(method='loess', se=TRUE, color='gray', alpha=0.1, size=0.7) + 
         geom_smooth(method='lm', se=FALSE, color='gray') + theme_minimal() + theme(legend.position='none') + 
         ggtitle('Mean expression vs model probability by Module'))

rm(plot_data2)

Probability and SD of level of expression

There is also a positive relation between the standard deviation of a gene and its regression score, the model could be capturing this characteristic of the genes to make the prediction, and could be introducing bias

plot_data %>% filter(sdExpr<0.5) %>% ggplot(aes(sdExpr, prob)) + geom_point(alpha=0.1, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.2) + 
              geom_smooth(method='lm', color='#999999', se=FALSE, alpha=1) + 
              theme_minimal() + ggtitle('SD vs model probability by gene')

This approximation curve looks like the opposite of the trend found between mean/sd and model scores

plot_data %>% ggplot(aes(meanExpr, sdExpr)) + geom_point(alpha=0.1, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.3) + 
              geom_smooth(method='lm', color='#999999', se=FALSE, alpha=1) + 
              scale_x_log10() + scale_y_log10() +
              theme_minimal() + ggtitle('Mean expression vs SD by gene')

Probability and lfc

There is a relation between probability and lfc, so it IS capturing a bit of true information (because lfc and mean expression were negatively correlated and it still has a positive relation in the model)

• It seems to capture the signal from under-expressed genes better than the over-expressed ones

plot_data = negative_set %>% mutate(ID=rownames(negative_set)) %>% 
            left_join(DE_info %>% mutate(ID=rownames(DE_info)), by='ID')

plot_data %>% filter(abs(log2FoldChange)<10) %>%
              ggplot(aes(log2FoldChange, prob)) + geom_point(alpha=0.1, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.3) + 
              theme_minimal() + ggtitle('lfc vs model probability by gene')

Conclusion

The model is capturing the mean level of expression of the genes (indirectly through module memberhsip), which is a strong bias found in the SFARI scores, but it seems to be capturing a bit of true biological signal as well (based on the GS and the log fold change plots)

---

Saving results

save(train_set, test_set, negative_set, fit, dataset, file='./../Data/LR_model.RData')

---

Session info

sessionInfo()

## R version 3.6.3 (2020-02-29)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB.UTF-8        LC_COLLATE=en_GB.UTF-8    
##  [5] LC_MONETARY=en_GB.UTF-8    LC_MESSAGES=en_GB.UTF-8   
##  [7] LC_PAPER=en_GB.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] expss_0.10.2       corrplot_0.84      car_3.0-7          carData_3.0-3     
##  [5] ROCR_1.0-7         gplots_3.0.3       Rtsne_0.15         biomaRt_2.40.5    
##  [9] RColorBrewer_1.1-2 gridExtra_2.3      viridis_0.5.1      viridisLite_0.3.0 
## [13] plotly_4.9.2       knitr_1.28         forcats_0.5.0      stringr_1.4.0     
## [17] dplyr_0.8.5        purrr_0.3.3        readr_1.3.1        tidyr_1.0.2       
## [21] tibble_3.0.0       ggplot2_3.3.0      tidyverse_1.3.0   
## 
## loaded via a namespace (and not attached):
##   [1] readxl_1.3.1                backports_1.1.5            
##   [3] Hmisc_4.4-0                 lazyeval_0.2.2             
##   [5] splines_3.6.3               BiocParallel_1.18.1        
##   [7] crosstalk_1.1.0.1           GenomeInfoDb_1.20.0        
##   [9] digest_0.6.25               htmltools_0.4.0            
##  [11] gdata_2.18.0                fansi_0.4.1                
##  [13] magrittr_1.5                checkmate_2.0.0            
##  [15] memoise_1.1.0               cluster_2.1.0              
##  [17] openxlsx_4.1.4              annotate_1.62.0            
##  [19] modelr_0.1.6                matrixStats_0.56.0         
##  [21] prettyunits_1.1.1           jpeg_0.1-8.1               
##  [23] colorspace_1.4-1            blob_1.2.1                 
##  [25] rvest_0.3.5                 haven_2.2.0                
##  [27] xfun_0.12                   crayon_1.3.4               
##  [29] RCurl_1.98-1.1              jsonlite_1.6.1             
##  [31] genefilter_1.66.0           survival_3.1-11            
##  [33] glue_1.3.2                  gtable_0.3.0               
##  [35] zlibbioc_1.30.0             XVector_0.24.0             
##  [37] DelayedArray_0.10.0         BiocGenerics_0.30.0        
##  [39] abind_1.4-5                 scales_1.1.0               
##  [41] DBI_1.1.0                   miniUI_0.1.1.1             
##  [43] Rcpp_1.0.4                  xtable_1.8-4               
##  [45] progress_1.2.2              htmlTable_1.13.3           
##  [47] foreign_0.8-75              bit_1.1-15.2               
##  [49] Formula_1.2-3               stats4_3.6.3               
##  [51] htmlwidgets_1.5.1           httr_1.4.1                 
##  [53] acepack_1.4.1               ellipsis_0.3.0             
##  [55] pkgconfig_2.0.3             XML_3.99-0.3               
##  [57] farver_2.0.3                nnet_7.3-13                
##  [59] dbplyr_1.4.2                locfit_1.5-9.4             
##  [61] later_1.0.0                 tidyselect_1.0.0           
##  [63] labeling_0.3                rlang_0.4.5                
##  [65] AnnotationDbi_1.46.1        munsell_0.5.0              
##  [67] cellranger_1.1.0            tools_3.6.3                
##  [69] cli_2.0.2                   generics_0.0.2             
##  [71] RSQLite_2.2.0               broom_0.5.5                
##  [73] fastmap_1.0.1               evaluate_0.14              
##  [75] yaml_2.2.1                  bit64_0.9-7                
##  [77] fs_1.4.0                    zip_2.0.4                  
##  [79] caTools_1.18.0              nlme_3.1-144               
##  [81] mime_0.9                    ggExtra_0.9                
##  [83] xml2_1.2.5                  compiler_3.6.3             
##  [85] rstudioapi_0.11             curl_4.3                   
##  [87] png_0.1-7                   reprex_0.3.0               
##  [89] geneplotter_1.62.0          stringi_1.4.6              
##  [91] highr_0.8                   lattice_0.20-40            
##  [93] Matrix_1.2-18               vctrs_0.2.4                
##  [95] pillar_1.4.3                lifecycle_0.2.0            
##  [97] data.table_1.12.8           bitops_1.0-6               
##  [99] httpuv_1.5.2                GenomicRanges_1.36.1       
## [101] R6_2.4.1                    latticeExtra_0.6-29        
## [103] promises_1.1.0              KernSmooth_2.23-16         
## [105] rio_0.5.16                  IRanges_2.18.3             
## [107] gtools_3.8.2                assertthat_0.2.1           
## [109] SummarizedExperiment_1.14.1 DESeq2_1.24.0              
## [111] withr_2.1.2                 S4Vectors_0.22.1           
## [113] GenomeInfoDbData_1.2.1      mgcv_1.8-31                
## [115] parallel_3.6.3              hms_0.5.3                  
## [117] grid_3.6.3                  rpart_4.1-15               
## [119] rmarkdown_2.1               Biobase_2.44.0             
## [121] shiny_1.4.0.2               lubridate_1.7.4            
## [123] base64enc_0.1-3